An automobile tire has a volume of 0.0150 m^3 on a cold day when the temperature of the air in t

step 1 Hi everyone here it is given an automobile tire having the volume 0 .015 meter cube on cold day

An automobile tire has a volume of 0.0150 m^3 on a cold day...

An automobile tire has a volume of 0.0150 $\mathrm{m}^{3}$ on a cold day when the temperature of the air in the tire is $5.0^{\circ} \mathrm{C}$ and atmospheric pressure is 1.02 $\mathrm{atm}$ . Under these conditions the gauge pressure is measured to be 1.70 atm (about 25 Ib $/$ in. $^{2}$.) After the car is driven on the highway for 30 $\mathrm{min}$ , the temperature of the air in the tires has risen to $45.0^{\circ} \mathrm{C}$ and the volume has risen to 0.0159 $\mathrm{in}^{3} .$ Whatthen is the gauge pressure?

Key Concepts

Ideal Gas Law

The ideal gas law, expressed as PV = nRT, relates the pressure (P), volume (V), the amount of gas (n), and the absolute temperature (T) through the gas constant (R). This principle allows one to predict how changes in temperature and volume will affect the pressure if the amount of gas remains constant, which is critical when analyzing the behavior of gases in systems like automobile tires as they heat up and expand.

Gauge Pressure vs. Absolute Pressure

Gauge pressure is the pressure measured relative to the ambient atmospheric pressure, meaning it is the amount by which the total (absolute) pressure exceeds the atmospheric pressure. Understanding this distinction is important because many practical pressure measurements are provided as gauge values, and converting between gauge and absolute pressures is necessary when applying the ideal gas law, which requires absolute temperatures and pressures.

Temperature Conversion

Temperature conversion, particularly converting from Celsius to Kelvin, is a key step when using the ideal gas law, as the law requires temperatures to be in an absolute scale. The Kelvin scale is used because it begins at absolute zero, ensuring proportional relationships in thermodynamic calculations such as predicting pressure changes due to heating or cooling.

Volume Change Due to Thermal Expansion

The volume change of a container, such as a tire, is another important concept when dealing with gases, as the volume may vary with changes in temperature and pressure. Recognizing how the expansion of the tire's material affects the internal volume is crucial in setting up the correct thermodynamic model to calculate the resultant pressure after heating.

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